Q23. - Algebra 1

Question

Find the next three terms in each geometric sequence.

$1000,500,250\mathrm{.....}$

Step-by-Step Solution

Answer

The next three terms in the geometric sequence are $125,62.5,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}31.25}$.

Step 1. State the concept used.

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. 

For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3.

The ration of 2^nd term and first term is called the common ratio. 

Step 2. Find the common difference.

The sequence: $1000,500,250\mathrm{.....}$

Each term in a geometric sequence can be expressed in terms of the first term $a_1$ and the common ratio r. Since each succeeding term is formulated from one or more previous terms, this is a recursive formula.

First term $a_1=1000$ and the common difference is:

$$\begin{gathered} r=\frac{a_2}{a_1} \\ =\frac{500}{1000} \\ =\frac{1}{2} \end{gathered}$$. 

Step 3. Find the next three terms.

In order to calculate the next three terms of the geometric sequence substitute $n=4,5,6$ and 1000 for $a_1$ into the formula $a_n=a_1{r}^{n-1}$.

 

Terms	Symbol	In terms of $a_1$ and r	Numbers
Fourth term	$a_4$	$a_1\cdot r^3$	$1000\times {\left(\frac{1}{2}\right)}^3=1000\times \left(\frac{1}{8}\right)=125$
Fifth term	$a_5$	$a_1\cdot r^4$	$1000\times {\left(\frac{1}{2}\right)}^4=1000\times \left(\frac{1}{16}\right)=62.5$
Sixth term	$a_6$	$a_1\cdot r^5$	$1000\times {\left(\frac{1}{2}\right)}^5=1000\times \left(\frac{1}{32}\right)=\text{31.25}$

 

Thus, next three terms of the sequence $1000,500,250\mathrm{.....}$ are $125,62.5,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}31.25}$.